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This work of art was originally posted on MathForums.com by Ty Odle.
I believe I have answered the Collatz Conjecture (work shown below in writing and images). I should not have crossed out the work I did as it directly contributes to the answer, but it’s clearly visible.
My theory is that Collatz conjecture proves 0 to the 0 power equals -1, not 1, as previously theorized separate from the Collatz Conjecture. When plugging 0 in for “x” we are able to bypass the conjecture since 0 is an even number. That leads us to T(0) = 0/2 and since 0 is a complex number, the right side of the equation requires we apply 0 the imaginary value i^2–1. So now we have T(0)=i^2–1/2 – the value of the variable T is zero or undefined, so it must be simplified down to 0 to the 0 power. i^2 in the numerator is cancelled out by the common denominator /2. What are you left with? 0^0 = −1
It’s my belief mathematicians have been asking the wrong question all along; “to what end does the Collatz Conjecture go?” rather than “why does the conjecture happen?” With that being said I’ll leave you with the famous quote by the man himself..
"We cannot solve our problems with the same thinking we used when we created them." —Albert Einstein
I'm sorry, what? 0=(i^2−1)/2 is correct [sic], because i^2=−1. 0=i/2 is absolutely wrong and you should have realised that once you got to 0=−1.
0 = i/2 is absolutely wrong, you are correct. I should have mentioned that factor did not fall into the equation, so that part should stay crossed out. When I got to 0 = −1 I forgot to factor in T, so T(0) = −1 equals 0^0 = −1
More criticism and feedback is much appreciated.
1. Nothing you have written here has anything to do with the Collatz conjecture.
2. Everything you have written here is utter nonsense.
3. 0^0 has no natural definition. But 0^0=−1 is not a definition which is even sensible.
Is it nonsense because it’s incomprehensible?
I think you’re reading my nonsense and thinking it factors into the equation… when it doesn’t. Most of the stuff (if not all the stuff) I wrote on the back of the paper is nonsense. But if you have any imagination whatsoever you can clearly read in between the lines and see what I’m getting at. Did you read the paragraph before the photos or just look at the photos?
No, he thinks it's nonsense. So do I. The whole *point* of a proof is that you don't have to use your imagination to see what is going on. You are trying to show everyone else.
I'll repeat a question already asked: What does this have to do with the Collatz conjecture? Can you show what happens to the orbit of 368468468498748484414548?
-Dan
“The Collatz conjecture in mathematics asks whether repeating two simple arithmetic operations will eventually transform every positive integer into one.” This quote is absolute bs. The Collatz conjecture asks the wrong question. The real question is WHY does this happen and I’m giving you guys the answer. 0^0 = −1 prove me wrong please
In this thread I am stating to the world that 0^0 = −1 and not 0^0 = 1. The Collatz conjecture is what proves this notion.
I'm not breaking anything! I'm just wondering why the word "orbit" isn't even in your proof when that's what the conjecture is about.
And
0^0≠−1
First you say T(0) = 0. Then you say
T(0)=(i^2−1)/2 ⟹ T(0)=−1
Since
0≠−1
you have shown that something is wrong.
What's wrong is that
(i^2−1)/2=−1
, not 0 so you can't put
(i^2−1)/2=0
. What you must have meant is
(−(i^2)−1)/2=0
which does lead to 0 = 0. But then you are stuck and your “proof” gets nowhere..
And before you start getting short with us, and you are apparently starting to get steamed, look at post #4.
-Dan
Addendum: In post #9 you say
“The Collatz conjecture in mathematics asks whether repeating two simple arithmetic operations will eventually transform every positive integer into one.” This quote is absolute bs.
That means you are saying that the conjecture is false and you’ve stopped talking about it!
I’m going to quote you Dan, “What's wrong is that (i^2−1)/2=−1”; this is only supporting my case and I want to thank you for pointing it out. You forgot T(0) but I’ll forgive you. So T(0) = (i^2-1)/2 then T(0) = −1 then 0^0 = −1.
Picture 4 people in a room together; 3 of which are wearing shirts with the numbers: −1, 0 , and 1. The 4th person (not wearing a number shirt) asks the 3 people “who’s hungry, who needs food?”... the person wearing the 1 shirt says “me!”, leaves the room to go get food. The person wearing the 0 shirt doesn’t say anything, so they sit there quietly and obviously they aren’t going to eat. The person wearing the −1 shirt says “no” so they do not eat either. 0 and −1 have met the same fate, neither will eat. Thus they are treated as equals.
So the person wearing the −1 shirt is obviously [negative], 0 shirt is neutral, and 1 shirt? You guessed it, [positive]. Well where does that leave the 4th person? They remained neutral and did not answer their own question. Proving them to be a 0 as well, pairing them together with the other 0. Now that the 4th person has been included, “the power”, you now have yourself an equation. 0^0 = −1
Might I add that since the 4th person is “the power”, they can make the choice whether or not to feed themselves. Their decision is unknown, their decision is a variable. That’s why their value as “the power” can also be interpreted as a variable. Hence why T(0) is simplified to 0^0.
You asked for my feedback. Here it is:
Scrap this idea. There's no consistent thread of logic throughout your argument. If you want to prove something about Collatz’ Conjecture, I suggest you spend at least 10 years of your life actually studying what other mathematicians have done with it.
Thank you. I’d rather not waste 10 years of my life learning from other’s mistakes. I’ll take my chances making mistakes on my own.
Good. Because you’ve made several. That’s how you learn. Start over and make sure your next proof actually says something about the Collatz conjecture.
The Collatz Conjecture proves 0^0 = −1
Thank you. I’d rather not waste 10 years of my life learning from other’s mistakes. I’ll take my chances making mistakes on my own.
That’s a shame. Most of the peer reviewed papers you’ll read have very, very minor mistakes, if any. You’d do well to read some and learn how they go about proving difficult problems.
So apparently 0^0 = −1 is incomprehensible to most, if not all people reading this thread... understandably. We’ve been taught that 0^0 = 1, calculators are programmed to calculate 0^0 = 1, and yet you all refuse to question the reality of this false equation. I stand firm in my belief that 0^0 = −1 and it is proven in the Collatz conjecture by plugging 0 in for x. I’ll leave it you all to critique or debate thread and I will no longer be commenting.
So much for wanting feedback. What you actually wanted was to tell us we’re wrong.
All right. Let's take this from the top.
The Collatz conjecture is that given a positive integer seed number a we define the function
T_n+1(a) = (3T_n(a) + 1)/2 if T_n(a) is odd > = T_n(a)/2 if T_n(a) is even
where
T1(a)=a
, that eventually
Tm(a)=1
for some integer m for any possible a.
You start by using a seed of 0, which is outside the domain. But, hey, it might prove to be useful, so let's do it!
T1(0)=0 ⟹ T2(0)=0
and so on. So this series never becomes 1. Which is okay because the seed is supposed to be a positive integer and here it isn't.
Now what? Suddenly you come up with
(i^2−1)/2=−1
and you say that
T2(0)=(i^2−1)/2
. *Why do you say that?*
Then you put both together and come up with
T2(0) = 0 = −1
Your proof ends here because we have a paradox. 0 does not equal −1 in any Mathematical system I have ever heard of. You would have to redefine what equality means and you didn’t say anything about doing that. What this line says is that you have made a mistake.
And that’s as far as this is going to go.
—Dan
since 0 is an even number we bypass the conjecture???
there is a cycle of 1 even value to it self there you know ...
plus there are overall 5 known cycles over the integers!
which are:
0 → 0 ... 1 → 4 → 2 → 1 ... −1 → −2 → −1 ... −5 → −14 → −7 → −20 → −10 → −5 ... −17 → −50 → −25 → −74 → −37 → −110 → −55 → −164 → −82 → −41 → −122 → −61 → −182 → −91 → −272 → −136 → −68 → −34 → −17 ...
since 0 has no value we must apply value i^2−1
why ???
Correction on Line 1. 3x − 1 should be 3x + 1.
The cycles of integers you speak of are obsolete.
0 has no value so we must assign it value to finish the problem. The only value we can assign it is the imaginary value of i^2−1
First:
i^2 − 1 = −2
is not imaginary. It’s an integer.
Second: You can't cancel like that. To cancel you have to have a common factor in the numerator and denominator.
(i^2 − 1)/2 = (−1−1)/2 = −2/2 = −1
You can’t just cancel the i^2 and the 2.
Third: Again, why are you assigning
(i^2 − 1)/2
for 0? It’s equal to −1, not 0.
Fourth: If you ever get 0 = −1 then you have a mistake somewhere. You can never say that this is true.
-Dan
Addendum: You said earlier others say that
0^0=1
They shouldn’t... that’s not a true statement, either.
I didn’t say you could cancel the i^2 and the 2, I said you can cancel i^2 and the /2.
I am assigning the 0 a value because it has none, the only applicable value would be the imaginary number formula i^2=−1. Since we already have an “=”, it can be dropped from the equation, leaving the only applicable integers from said formula, i^2−1.
0 is both a real and imaginary number. Imaginary numbers can be replaced by many imaginary expressions so long as “i” always equals 0. (i=0) We know “i” will always equal 0 no matter the case because 0 is in fact, an imaginary number.
I didn’t say you could cancel the i^2 and the 2, I said you can cancel i^2 and the /2.
Same problem. That's not a valid algebraic operation.
I am assigning the 0 a value because it has none, the only applicable value would be the imaginary number formula i^2=−1.
Why does 0 not have a value? And why does not having a value immediately imply that you need imaginary numbers?
0 is both a real and imaginary number. Imaginary numbers can be replaced by many imaginary expressions so long as “i” always equals 0. (i=0) We know “i” will always equal 0 no matter the case because 0 is in fact, an imaginary number.
Umm... no. You're right that 0 is both a real and complex (not imaginary [sic]) number, but i≠0.
I thought you said you weren’t replying to this thread any more?
You need to review some basic algebra. Let's look at this:
(x+1)/x
According to you this is
(<s>x</s>+1)/<s>x</s> = 1x
But we can put values in for x to show that this isn’t true in general (or at all.) Let x = 1:
(1+1)/1 = 2 ≠ 1
Now if you had something like
(x^2+x)/x = x(x+1)/x = <s>x</s>(x+1)/<s>x</s> = x+1
because x is a factor in both the numerator and denominator.
By definition
i^2=−1
If i = 0 we have
0^2 = 0 ≠ −1
So
i≠0
You can’t just change the value of something in the middle of the problem!
And even if i = 0,
(i^2−1)/2 = (0^2−1)/2 = −1/2
so you can’t replace 0 with it.
This is the third(?) time I’ve mentioned that you have a mistake because you are saying 0 = −1 and others have told you as well. This is a basic fact and you can’t get around it. If you can't accept that, then there is nothing more I can do.
I’m done with this.
—Dan
In writing you can see 0 = −1, but I never made that statement or confirmed it was factored in. What I did say about 0 = −1 is when I got to that part of the problem, I forgot I had to factor in “T”. Thus correcting myself T(0) = −1. This leads us directly to 0^0 = −1. I couldn’t put it any simpler. The fuss you all seek to invoke is “how the hell could he have solved the Collatz conjecture in only 4 steps while also defining the undefined 0^0?” It’s blowing your guys’ mind.
Okay, I can’t let this go on. I don’t care you did it in 4 lines. You didn’t actually do anything but make a mistake!! You even tossed off the original problem as “obvious” but never proved it.
I really don’t know what your problem is. Is there any justification of saying that 0 = −1? No!
Your “proof” doesn't prove anything, it is merely mentions the Collatz conjecture and never talks about it again. You never even use the work “orbit” which is crucial to the Collatz conjecture. And you have run into a step that is a contradiction, 0 = −1, and you don’t apparently have enough Mathematics background to understand that this is a fatal sign.
Learn some Math, tinker with it again, and come back when you have grown up enough to accept that someone else might be right and you wrong.
-Dan
Addendum: Oh, and one last thing. The next time you present a proof write it on paper that isn’t your grocery list. That doesn’t impress anyone.
Thus correcting myself T(0) = −1.
That’s still wrong, since T(0)=0.
The fuss you all seek to invoke is “how the hell could he have solved the Collatz conjecture in only 4 steps while also defining the undefined 0^0?” It’s blowing your guys’ mind.
That’s not what we’re saying. The only thing blowing my mind is how you could think that your work proves anything. It’s complete jibberish.
You keep saying the word “orbit” like it has any relevance to my theory whatsoever. In math we speak with numbers not words.
That's still wrong, since T(0)=0
That’s not what we’re saying. The only thing blowing my mind is how you could think that your work proves anything. It’s complete jibberish.
Plug 0 in for x in the Collatz conjecture and show me what you get.
What does the conjecture suggest? Does it mention x?
The conjecture in itself asks the wrong question “to what extent, by these rules” instead of “why does it happen.” That’s the first problem every close-minded commenter comes to this thread with. The conjecture is a barrier, 3x+1. To solve this problem we must go around this barrier and come back to it with a proper value not within the boundaries set by theorists on the conjecture. Plugging 0 in for x allows us to bypass the barrier (or conjecture, “if odd”) by using an even number “0”. Which leads us to where we want to go, T(0) = 0/2. The event horizon of my theory is this: the proper value we will always come back to it with after plugging in 0 for x is −1. Plug −1 into the whole idea behind the conjecture, “that it all leads back to +1” and you have yourself a zeroed out equation. A solved conjecture. The conjecture defines the value of 0 to the 0 power, −1.
side note: I appreciate the criticism and feedback as it has helped me better explain my theory to you all and to myself. But I don’t appreciate the hate mail on this thread. Every jab you guys take at me personally discredits your opinions on the matter, to me and to all. If you’re gonna come here and critique my theory please do it respectfully.
If you alter the conjecture, you’re no longer working on the original Collatz conjecture.
I have to disagree. The definition of conjecture is “an opinion or conclusion formed on the basis of incomplete information.” Incomplete information is unalterable.
Thanks to everyone’s help here on the thread, I have rewritten and published my theory in multiple areas of discussions online:
My theory is that Collatz conjecture proves 0 to the 0 power equals −1, not 1 or undefined, as previously theorized in arithmetic separate from the Collatz Conjecture. When plugging 0 in for “x” we are able to bypass the conjecture since 0 is an even number.
The conjecture in itself asks the wrong question “to what extent, by these rules” instead of “why does it happen.” That’s where close-mindedness leads to a problem when solving a conjecture. Some argue I am altering the conjecture, but the definition of a conjecture is “an opinion or conclusion formed on the basis of incomplete information.” To this I say: “incomplete information is unalterable.” The conjecture is a barrier, 3x+1. To solve this problem we must go around this barrier and come back to it with a proper value not within the boundaries set by theorists on the conjecture. Plugging 0 in for x allows us to bypass the barrier (or conjecture, “if odd”) by using an even number “0”. Which leads us to where we want to go, T(0) = 0/2. 0/2 has no assigned value so we must replace or assign value to 0 with the imaginary number i^2−1 and put it over 2, leaving us with T(0)=(i^2−1)/2, T(0)=−1, then 0^0=−1 The event horizon of my theory is this: the proper value we will always come back to it with after plugging in 0 for x is −1. Plug −1 into the whole idea behind the conjecture, “that it all leads back to +1” and you have yourself a zeroed out equation. A solved conjecture. The conjecture defines the value of 0 to the 0 power, −1.
I’ll paint a picture of my of my theory you in the setting of our three-dimensional world:
Let’s say 4 people are in a room together; 3 of which are wearing shirts with the numbers: −1, 0, and 1. The 4th person (not wearing a number shirt) asks the 3 people “who’s hungry, who needs food?”... the person wearing the 1 shirt says “me!”, leaves the room to go get food. The person wearing the 0 shirt doesn’t say anything, so they sit there quietly and obviously they aren’t going to eat. The person wearing the −1 shirt says “no” so they do not eat either. 0 and −1 have met the same fate, neither will eat. Thus they are treated as equals. So the person wearing the −1 shirt is obviously [negative], 0 shirt is neutral, and 1 shirt? You guessed it, [positive]. Well where does that leave the 4th person? They remained neutral and did not answer their own question. Proving them to be a 0 as well, pairing them together with the other 0. Now that the 4th person has been included, “the power”, you now have yourself an equation. 0^0 = −1
The Collatz conjecture is a question asking “what” instead of “why” and I gave you the answer as to “why” while also defining the previously undefined value of 0^0, essentially killing two birds with one stone.
"We cannot solve our problems with the same thinking we used when we created them." —Albert Einstein
Every jab you guys take at me personally discredits your opinions on the matter, to me and to all. If you’re gonna come here and critique my theory please do it respectfully.
What else do you expect? You come in to a forum with some reasonably good mathematicians. A forum that has a long history of crank posts that everyone is sick and tired of.
Then you ask for feedback on your idea and the best any of us can do is “it’s wrong”. Not because we couldn’t be bothered, but because your attempts are so far separated from mathematics that nobody here can make heads or tails of it. Then you refuse to acknowledge other people’s ideas and reveal that you don’t trust anybody with maths knowledge, so by extension you don’t trust us. And you expect us to be respectful?
https://www.reddit.com/r/math/comments/iagseh/whats_the_difference_between_a_crank_and_a/
You come up with a novel idea. How do mathematicians tell what's cranky and not?
There are two parts to crankery:
* Bad mathematics. This is usually:
** Stuff that contradicts existing mathematics, like claiming that pi is rational;
** Stuff that isn't internally consistent, like letting 1/0 = w, but also saying that the field axioms still hold (this leads to a contradiction);
** Proofs that flat-out don’t work, like trying to prove Collatz by using a probabilistic argument;
** Unclear explanations, or a lack of definitions, like making continual reference to a “Todd function” which has never been described in the literature;
** Complete nonsense Not Even Wrong word salad, like Time Cube.
* A lack of humility, especially when told that their mathematics is bad. This can take the form of:
** Naming their own “contributions” after themselves (mathematicians tend to only have their work named after them once people use their result frequently enough to need to name it);
** Expecting other people to do the legwork of actually checking if their idea holds water;
** Getting unreasonably angry when people point out why their mathematics is bad;
** Insulting other people for being stupid and “not understanding their work” (bonus points if they didn’t define their terms properly);
** Insisting that mainstream mathematics is inconsistent without a properly solid proof (bonus points if their mathematics education is insufficient or if they're self-taught)
** Insisting that mainstream mathematicians are all “brainwashed” or “sheep”, or that there’s a conspiracy to hide the truth, or that they’re being persecuted for telling the truth, etc.
The first part alone isn’t necessarily bad unless it’s word salad; this sort of thing can often come from curious highschoolers or enthusiasts, and should be corrected and nurtured. It’s the second part that makes mathematicians go “Ugh, great, a crank”.
IMO, finitism itself isn’t necessarily bad mathematics, because it’s possible to work in ZF-Infinity (and to do so very honestly as long as you state so up-front). Hell, if you can inform me of a consistent foundation for ultrafinitism, then I’ll happily grant it as not-crankery. However, every single finitist I’ve seen or heard of tries to disprove the Axiom of Infinity, or claim that it’s somehow inconsistent or nonsense (without actually providing a sound proof), and then flings shit at the mathematical community for quite rightly calling them out on this.
Similarly, I’m of the opinion that Mochizuki and his cohort are cranks. They’ve been pointed to a specific hole in IUT, and given a counterexample to said hole, and so far they seem to keep claiming “Western bias”, or that mathematicians “aren’t trying to understand the proof”, or that the problem “doesn’t exist”, instead of actually addressing the problem.
As for paraconsistent logics, I’m willing to make an exception, but that’s because I haven’t looked into this well enough to form any opinion on whether its foundations are solid. I hope they are.
A quote from Paraconsistent Logic (Stanford Encyclopedia of Philosophy)
2.4 Arithmetic and Godël’s Theorem
“One version of Gödel’s first incompleteness theorem states that for any consistent axiomatic theory of arithmetic, which can be recognised to be sound, there will be an arithmetic truth—viz., its Gödel sentence—not provable in it, but which can be established as true by intuitively correct reasoning. The heart of Gödel’s theorem is, in fact, a paradox that concerns the sentence, G, ‘This sentence is not provable’. If G is provable, then it is true and so not provable. Thus G is proved. Hence G is true and so unprovable. If an underlying paraconsistent logic is used to formalise the arithmetic, and the theory therefore allowed to be inconsistent, the Gödel sentence may well be provable in the theory (essentially by the above reasoning). So a paraconsistent approach to arithmetic overcomes the limitations of arithmetic that are supposed (by many) to follow from Gödel’s theorem.”
How much of that excerpt do you actually understand?
i am going to say it once (here in this thread — not on other threads)
the collatz conjecture is all about primes and the ln3/ln2 line with a floor function!
btw you are wrong Ty Odle and I am saying this calmly and nicely;
I am sorry that it is not what you thought, but what you are saying is simply wrong
and believe me I am open-minded as one can be.
I have to disagree. The definition of conjecture is “an opinion or conclusion formed on the basis of incomplete information.”
That is the first definition suggested by Google, but for mathematical purposes, a conjecture is simply an unproven mathematical statement. The Collatz conjecture is a precise, but unproven, mathematical statement about integers greater than zero. Many variations have been considered, some of which are easily proved, but considering variations is different from considering the original conjecture.
How much of that excerpt do you actually understand?
Every bit of it. It implies my theory is unquestionably true.
That is the first definition suggested by Google, but for mathematical purposes, a conjecture is simply an unproven mathematical statement. The Collatz conjecture is a precise, but unproven, mathematical statement about integers greater than zero. Many variations have been considered, some of which are easily proved, but considering variations is different from considering the original conjecture.
“unproven, mathematical statement?”
“incomplete, mathematical information?” Sounds like “potato, potato” to me. So since no one can argue over the proof I’ve given, we have to resort to arguing over a definition?
One thing at a time. Proofs usually start with definitions, so that it's completely clear what you are doing. Also, “incomplete” is non-specific as to what is missing, whereas “unproven” means that a proof of a particular mathematical statement is missing.
Proof/definition is 0^0 = −1
There’s nothing interesting after this point.
This is a work of art, so I had to preserve it.
